Definition 5.18.3670832106E8E

Definition 5.18.3.670832106E8E Let $D$ be an effective $\mathbb {Q}$-divisor¹ on $X$. We say $\varphi \in \mathrm{QPSH}(X)$ has log singularities (along $D$) on $X$ if for each $x\in X$, there is an open neighborhood $U$ of $x$ such that

1. 05DF50833BB0

   $D|_ U$ has finitely many irreducible components and can be written as

   \[ D|_ U=\sum _{i=1}^ N a_ i D_ i \]

   with $D_ i$ prime divisors on $U$, $a_ i\in \mathbb {Q}_{>0}$ and $s_ i$ holomorphic functions on $U$ defining $D_ i$, and

2. B8CC79988714

   we have

   C6AC1C84A3B3 5.18.4

   \begin{equation} \varphi |_ U=\sum _{i=1}^ N a_ i\log |s_ i|^2+R, \end{equation}

   where $R$ is a bounded function on $U$.